Properties

Label 2-7200-1.1-c1-0-69
Degree $2$
Conductor $7200$
Sign $-1$
Analytic cond. $57.4922$
Root an. cond. $7.58236$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7-s + 13-s − 3·19-s + 4·23-s − 4·29-s + 7·31-s − 6·37-s − 6·41-s − 9·43-s + 6·47-s − 6·49-s − 2·53-s + 10·59-s − 61-s + 3·67-s − 14·71-s + 10·73-s − 8·79-s + 18·83-s − 91-s − 3·97-s + 6·101-s − 8·103-s + 2·107-s − 15·109-s + 12·113-s + ⋯
L(s)  = 1  − 0.377·7-s + 0.277·13-s − 0.688·19-s + 0.834·23-s − 0.742·29-s + 1.25·31-s − 0.986·37-s − 0.937·41-s − 1.37·43-s + 0.875·47-s − 6/7·49-s − 0.274·53-s + 1.30·59-s − 0.128·61-s + 0.366·67-s − 1.66·71-s + 1.17·73-s − 0.900·79-s + 1.97·83-s − 0.104·91-s − 0.304·97-s + 0.597·101-s − 0.788·103-s + 0.193·107-s − 1.43·109-s + 1.12·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7200\)    =    \(2^{5} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(57.4922\)
Root analytic conductor: \(7.58236\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 7200,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 3 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 9 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 - 3 T + p T^{2} \)
71 \( 1 + 14 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 18 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 3 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.55450369771688035071714991592, −6.73472876033950905183980088011, −6.37095247036193255421476502980, −5.39830694538726179521354814029, −4.80234809036820990737382481889, −3.86244208296568369791086854386, −3.21026016842139130510207246795, −2.28967252494869327056409312340, −1.28329535957897760021317649668, 0, 1.28329535957897760021317649668, 2.28967252494869327056409312340, 3.21026016842139130510207246795, 3.86244208296568369791086854386, 4.80234809036820990737382481889, 5.39830694538726179521354814029, 6.37095247036193255421476502980, 6.73472876033950905183980088011, 7.55450369771688035071714991592

Graph of the $Z$-function along the critical line